Methods for Estimating Modal Bandwidth Spectral Dependence

ABSTRACT

Methods for estimating the Effective Modal Bandwidth (EMB) of laser optimized Multimode Fiber (MMF) at a specified wavelength, λS, based on the measured EMB at a first reference measurement wavelength, λM. In these methods the Differential Mode Delay (DMD) of a MMF is measured and the Effective Modal Bandwidth (EMB) is computed at a first measurement wavelength. By extracting signal features such as centroids, peak power, pulse widths, and skews, as described in this disclosure, the EMB can be estimated at a second specified wavelength with different degrees of accuracy. The first method estimates the EMB at the second specified wavelength based on measurements at the reference wavelength. The second method predicts if the EMB at the second specified wavelength is equal or greater than a specified bandwidth limit.

CROSS REFERENCES TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.62/407,695, filed Oct. 13, 2016, the subject matter of which is herebyincorporated by reference in its entirety.

BACKGROUND OF THE INVENTION

The present invention relates in general to the field of optical fibersand more specifically, to multimode fibers (MMF) designed for operationat multiple wavelengths. The present invention also relates to the fieldof modeling, designing, production, sorting and testing of MMFs. Morespecifically it relates to the estimation of the MMF EMB at multiplewavelengths.

The invention is also related to modal and chromatic dispersioncompensation in Vertical Cavity Surface Emitting Laser (VCSEL) based MMFchannels [1]. The methods described here can provide an estimation ofthe skew in radial DMD pulse waveforms (tilt) at different wavelengthswhich is critical in the field of modal-chromatic dispersioncompensation.

The need for higher bandwidth has been mainly driven by the increasingdemand for high-speed backbone data aggregation fueled by videotransmission, server applications, virtualization, and other emergingdata services. Cost, power consumption, and reliability advantages havefavored the predominance of short and intermediate reach opticalchannels employing transmitters utilizing VCSELs operating at 850 nmover MMF. MMF is currently utilized in more than 85% of datacenterinstallations, and has a larger core diameter than single-mode fiber(SMF), which reduces connection losses, relaxes alignment tolerances,and reduces connectorization cost.

Recently, new modulation technologies for VCSEL-MMF channels such asPAM-4, and Short Wavelength Division Multiplexing (SWDM) [SWDMalliance], has been proposed in order to increase the data rates.Standards organizations, including the Institute of Electrical andElectronics Engineers (IEEE) working group 802.3cd and the T11 TechnicalCommittee within the International Committee for Information TechnologyStandards (INCITS) PI-7, are already working on new applications forPAM-4 for optical serial rates over 50 Gb/s per wavelength.

The SWDM concept is similar to the Coarse Wavelength DivisionMultiplexing (CWDM), already used for SMF channels operating in the 1310nm spectral region. SWDM requires the specification of the minimum EMBat the wavelengths limits of the operating spectrum (e.g. 850 nm and 953nm).

The EMB is computed from DMD pulse measurements. The DMD test method,specified within standards organizations [2], describes a procedure forlaunching a temporally short and spectrally narrow pulse (referencepulse) from a SMF into the core of a MMF at several radial offsets [5].After propagating through the MMF under test, the pulses are received bya fast photodetector which captures all the MMF core power. The EMB isestimated by the Fourier domain deconvolution of the input pulse from aweighted sum of the received signals for each radial offset launch. Theset of weight values utilized in the computation belong to a set of tenrepresentative VCSELs described elsewhere [2]. Due to the testcomplexity, it is time consuming and the equipment required to performthe test is expensive; EMB test requirements for multiple wavelengthswill significantly increase testing time and consequently, increasefiber cost. A method to estimate the EMB from measurements at a singlewavelength would therefore, reduce testing time and cost. The challengesto achieve such a method are described below.

FIG. 1, shows a simulation of EMB vs wavelength 100 for a MMF fibercompliant to the OM4 standard. In this figure, we show the EMB 105 has apeak value at λ_(P) 120. The labels 115 and 125 show the measured andpredicted wavelengths, λ_(M) and λ_(S), respectively. The range 110shows the spectral window in which the fiber can maintain an EMB higherthan a specified value, i.e. 4700 MHz·km for OM4.

In principle, based on MMF theory, when all the physical parameters ofthe fiber are known (i.e. dimensions, refractive profile, dopant typesand content), the EMB at λ_(S) can be predicted from the EMB value atλ_(M). In practice however, variations in the refractive index designand dopant content during the preform fabrication process producechanges in 100 which prevent the estimation of the EMB at λ_(S). FIG. 2shows simulated MMFs with identical EMB at λ_(M) 200, but different EMBspectral dependence. Peaks 205, 210, 205 are different and uncorrelatedwith 200. Moreover, since the spectral windows 220, 225 and 230 aredifferent, an estimation of the EMB at λ_(S) 240 is not possible.

Shown in FIG. 3. is the EMB at 850 nm and 953 nm for a large number ofsimulated fibers, represented using rectangle markers 300 with randomvariations in their refractive index core. The horizontal and verticalaxes of this figure represents the EMB at λ_(M)=850 nm vs. EMB λ_(S)=953nm respectively. A subset of these fibers that meet the TIA-492AAAD OM4EMB specification are represented by diamonds markers 305. This figureshows the lack of correlation among EMBs at 850 nm and 953 nm. Forexample, in 310, a measured fiber with EMB=6000 MHz·km at 850 nm canhave any value from 1500 to 3000 MHz·km at 953 nm. Conversely, 315 showsthat a MMF with EMB=2000 MHz·km at 953 nm can have any value from 200 to15000 MHz·km at 850 nm. This simulation, which was extended for a largerange of wavelengths from 800 nm to 1100 nm, clearly shows that there isno direct relationship between the fiber's EMB at a specifiedwavelength, λ_(S), and the EMB at a measured wavelength, λ_(M), whenλ_(S)≠λ_(M).

A method that enables the prediction of the EMB at an arbitrarywavelength based on measurements at another wavelength is desirable toreduce testing time and cost of a MMF.

SUMMARY OF THE INVENTION

Methods for estimating the Effective Modal Bandwidth (EMB) of laseroptimized Multimode Fiber (MMF) at a specified wavelength, λ_(S), basedon the measured EMB at a first reference measurement wavelength, λ_(M).In these methods the Differential Mode Delay (DMD) of a MMF is measuredand the Effective Modal Bandwidth (EMB) is computed at a firstmeasurement wavelength. By extracting signal features such as centroids,peak power, pulse widths, and skews, as described in this disclosure,the EMB can be estimated at a second specified wavelength with differentdegrees of accuracy. The first method estimates the EMB at the secondspecified wavelength based on measurements at the reference wavelength.The second method predicts if the EMB at the second specified wavelengthis equal or greater than a specified bandwidth limit.

DETAILED DESCRIPTION OF THE INVENTION

The present invention discloses novel methods to estimate the EMB of aMMF at a desired wavelength, from measurements performed at anotherwavelength. The first method, Method 1, can be used to predict the EMBat an arbitrary wavelength, λ_(S), based on an EMB measurement at adifferent wavelength, λ_(M). The second method can be used to evaluateif the EMB at an arbitrary wavelength, λ_(S), is equal of greater than aminimum specified threshold. Each method provides different degree ofcomplexity and accuracy.

These methods can be used for the design and manufacturing processes ofMMF that have a core and a cladding where the index of refraction of thecladding is less than that of the core. The core has a gradient index ofrefraction which varies from a peak value at the center of the core to aminimum value at the cladding interface following a predominantalpha-profile function to minimize modal dispersion [JLT 2012].Refractive index profiles for two types of MMF are shown in FIGS. 4 and5. In FIG. 4 a traditional MMF refractive index profile is shown. Theprofile 400 does not present any abrupt discontinuity inside the core orinside the cladding. The propagating mode groups of this fiber are shownin 410. In FIG. 5 the refractive index profile 500 abruptly changes inthe cladding due to the refractive index trench 520 introduced toprovide lower bending loss. Labels 510 and 515 shows some of thepropagating and leaking mode groups respectively.

Waveguide theory for alpha-profile fibers has been well developed [ref].The theory can enable the modeling of fiber DMD behavior over a broadrange of wavelengths, when the profiles and dopants concentrations areknown. In practice however, due to manufacturing variations the designed“optimum” refractive index profile is distorted deterministically andrandomly. Very small alterations in 400 or 500, basically change the waythe mode groups 410, 510 interact with the variations in refractiveindex, which destroys or reduces the correlations among DMDs atdifferent wavelengths as it was showed in FIG. 3.

Method 1

This method, can be used to predict the EMB at an arbitrary wavelength,λ_(S), based on an EMB measurement at a different wavelength, λ_(M). Themethod was developed based on the inventors' realization that in orderto increase the correlation among EMB measurements at λ_(M), and asecond wavelength, λ_(S), a new approach that fully utilizes theinformation provided by the measured DMD waveforms is required. Themethod proposed here uses the DMD pulse waveform information at λ_(M),such as centroids, peak position, width, shapes, energy per radialoffset, and skews, to predict the EMB at a second wavelength.Statistical and signal processing techniques disclosed here, allow us toextract and utilize those parameters to distort the DMD pulse waveformsacquired at λ_(M), to predict the DMD pulse waveforms at λ_(S). Thismethod which requires a training of the algorithm, enables theprediction of EMBs at different wavelengths from one measurement. FIGS.6 and 7 show the block diagrams for the training and estimationprocesses respectively. For illustrative purposes, we use an example todescribe both methods.

Training for Method 1

In 600, the populations of TIA-492AAAD standards compliant OM4 fibersfrom two suppliers (A and B), which use different manufacturingprocesses are selected. It is understood that the population used hereis only an example and is not restricted to any specific number of fibersuppliers. In 602, we select a subset of 24 fibers from manufacturer Aand 12 from manufacturer B for training. In 604, the DMD of all fibersare measured at the first measurement wavelength, λ_(M)=850 nm, and thesecond specified wavelength, λ_(S), which in this example is taken to be953 nm. These measurements are stored in the array y(r, t, λ) foranalysis. FIGS. 8(a) and 8(b) show the DMD radial pulses for three MMFfrom each population at 850 nm (dark trace) and 953 nm (lighter trace).FIGS. 8(a) and 8(b) show that most of the fibers have similar DMD pulsesat low radial offset for both wavelengths. For population A, the DMDpulse shapes are very different at larger radial offset for the twowavelengths.

The EMBs computed from the measured DMD pulses for the A and Bpopulations at 850 nm and 953 nm are shown in FIG. 9. These measurementsagree with simulation results showed in FIG. 3, which indicates thatEMBs at different wavelengths are uncorrelated.

In step 606 of FIG. 6, the main features of the DMD pulses at eachwavelength are extracted. This process captures the main characteristicsrequired to describe the DMD pulses at each radial offset and eachwavelength for post-processing and analysis. As an illustrative example,here we extract the centroid, mean power, peak power value and position,and the root mean square (RMS) width. The centroid feature is computedusing,

$\begin{matrix}{C_{r,\lambda} = \frac{\sum\limits_{k}{t_{k}{y\left( {r,t_{k},\lambda} \right)}}}{\sum\limits_{k}{y\left( {r,t_{k},\lambda} \right)}}} & (1)\end{matrix}$

where r is the radial offset index that relates the position of thesingle-mode launch fiber to the MMF core center axis during the DMDmeasurement, t is the discrete length normalized temporal, k is the timeindex. The variable t and k are related to the number of temporalsamples simulated or acquired from the oscilloscope during DMDmeasurements at a given wavelength. The mean power is computed by,

$\begin{matrix}{{Ymean}_{r,\lambda} = \frac{\sum\limits_{k}{y\left( {r,t_{k},\lambda} \right)}}{\sum\limits_{k}t_{k\;}}} & (2)\end{matrix}$

The peak power is computed using,

Ypeak_(r,λ)=max_(t)(y(r,t,λ))   (3)

where max_(t)(.) is a function that finds the maximum of the DMD pulsesfor each radial offset and for each wavelength. The peak position iscomputed using.

P_(r,λ)=find_peak(y(r,t,λ))   (4)

where, find_peak is a function that finds the maximum value of the DMDpulses for each radial offset and for each wavelength. The RMS width ofthe pulse for each radial offset is computed,

$\begin{matrix}{W_{r,\lambda} = \sqrt{\frac{\sum\limits_{k}{\left( {t_{k} - C_{r,\lambda}} \right)^{2}{y\left( {r,t,\lambda} \right)}}}{\sum\limits_{k}{y\left( {r,t_{k},\lambda} \right)}} - \left( T_{REF} \right)^{2}}} & (5)\end{matrix}$

where T_(REF) is the RMS width of the reference pulse used for themeasurement. The features extracted from DMD measurements at λ_(M), areused to predict features at λ_(S), based on the model described inequations (6-8).

C _(r,λ) _(S) =(1+I _(C)(r))C _(r,λ) _(M) +F _(C)(λ_(M),λ_(S))G _(C)(r)  (6)

where C_(r, λ) _(S) , and C_(t,λ) _(M) represent the centroids perradial offset at λ_(M) and λ_(S), I_(C)(.,.),F_(C)(.,.),G_(C)(.)is theset of polynomial functions that describe the relationship betweencentroids at those wavelengths.

P _(r,λ) _(S) =(1+I _(P)(r))P _(r,λ) _(M) +F _(P)(λ_(M),λ_(S))G _(P)(r)  (7)

where P_(r,λ) _(S) , and P_(r,λ) _(M) represent the centroids per radialoffset at λ_(M) and λ_(S), I_(P)(.,.), F_(P)(.,.),G_(P)(.) is the set ofpolynomial functions that describe the relationship between peakpositions at those wavelengths.

W _(r,λ) _(S) =(1+I _(W)(r))W _(r,λ) _(M) +F _(W)(λ_(M),λ_(S))G _(W)(r)  (8)

where W_(r,λ) _(S) , and W_(r,λM) represent the centroids per radialoffset at λ_(M) and λ_(S), I_(W)(.,.),F,_(W)(.,.),G_(W)(.) is the set ofpolynomial functions that describe the relationship between widths atthose wavelengths.

The F(.,.) functions are solely dependent on the measured and targetedwavelength. These functions accommodate for chromatic effects in therefractive index and material. The G(.) functions are solely dependenton radial offsets and accommodate for relationships between the groupvelocity of DMD pulses at different radial offset in the fiber core. TheI(.) functions, dependent on the radial offset, accommodates for modetransition due to the change of wavelengths.

In step 608, the features extracted from the measured DMD pulses at thetwo wavelengths are used to find the coefficients of the polynomialfunctions described above (6-8). Standard curve fitting techniques areapplied as described in [3]. For the samples used in this example, FIGS.10(a) and (b), show the centroid features for 850 nm and 953 nm forradial offsets from 1 to 24 microns for the two fiber populations A(red) and B (blue). FIGS. 11(a) and (b), show the peak positions for 850nm and 953 nm for radial offsets from 1 to 24 microns for the two fiberpopulations. For these samples, F(850,953) was 16 ps/μm/km forpopulation A and 13.3 ps/μm/km for population B. The functionsG_(C)(.,.) and G_(P)(.,.) for a cubic polynomial curve fitting is shownin FIGS. 12(a) and 12(b) for fiber populations A and B respectively.Similarly, curves for the other features described above (1-5) areobtained.

In 610 the correlations among the features, i.e. the ones shown in FIGS.1-12 are evaluated. If they are higher than a determined threshold,e.g., 80%, the model is ready to use and the process end in 615. If not,in 612 the signal to noise ratio (SNR) of all DMD measurements areevaluated. If the noise of the measurements is higher than apre-determine threshold, the measurements need to be repeated. If theSNR is high, but the correlations are low, it is possible that thesamples do not represent the fiber population and a new set of sampleswill be required.

Method 2: Estimation Method

After training, the method for the DMD mapping and estimation, shown inFIG. 7, is ready to use. Here, we use the same example to describe theprocesses. In 700 the fibers that require EMB estimation are selected.In 702, the model described in (6-8) and the wavelengths (in this caseλ_(M)=850 nm , λ_(S),=953nm) are selected. In 704, the DMD at λ_(M) ismeasured. In 706 the features are extracted from the DMD pulse centroidsat λ_(M) using equations (1-5). In 708 the DMD pulses are estimated atλ_(S). Next, the model described in equations (6-8) is used to estimatethe features C_(r,λ) _(S) , P_(r,λ) _(S) ,W_(r,λ) _(S) , Ymax_(r,λ) _(S)Ymean_(r,λ) _(S) at λ_(S).

The parameter P_(r,λ) _(S) is used to reposition each of the DMD pulsesusing,

Y _(P)(r,t _(k),λ_(S))=y(r,t _(k)−(P _(r,λ) _(S) −P _(r,λ) _(M) , λ_(M))  (9)

where the y_(P)(.,.,.) array represents the estimated DMD pulses afterthe peak position correction.

The differences between the centroid and peak position are computed atboth wavelengths. The variation of these differences are computed asshown,

Δ=(C _(r,λ) _(S) −P _(r,λ) _(S) )−(C _(r,λ) _(M) −P _(r,λ) _(M) )   (10)

The parameter Δ is used to estimate the new width and skew of the DMDpulses at λ_(S). In the majority of cases, when, λ_(S)>λ_(M), the DMDpulse width tends to increase. Conversely, when λ_(S)<λ_(M), the widthtends to decrease. The changes in skew and width are corrected using alinear filter as shown,

$\begin{matrix}{{y_{W}\left( {r,t_{k},\lambda_{S}} \right)} = {\sum\limits_{i = 0}^{Ntaps}{A_{i}{y_{p}\left( {r,{t_{k} - {{Ki}\; \Delta}},\lambda_{S}} \right)}}}} & (11)\end{matrix}$

where y_(W)(.,.,.) represents the estimated DMD after equalization, i isthe equalizer tap index, Ntaps the number of taps, A_(i) represents thetap coefficient, K is a scaling factor.

For each fiber, the optimum values of Ntaps, A_(i), and K, are found bynumerically searching. The constraint conditions or equations for thissearch are the estimated mean, peak, and the values shown in table I.

TABLE I  $\frac{\sum\limits_{k}\; {y_{W}\left( {r,t_{k},\lambda_{S}} \right)}}{\sum\limits_{k}\; t_{k}} \leq {Ymean}_{r,\lambda_{S}}$max_(t) (y_(W) (r, t_(k), λ_(S))) ≤ Ypeak_(r,λ) _(S)$\sqrt{\frac{\sum\limits_{k}{\left( {t_{k} - C_{r,\lambda}} \right)^{2}{y_{W}\left( {r,t_{k},\lambda_{S}} \right)}}}{\sum\limits_{k}\; {y_{W}\left( {r,t_{k},\lambda_{S}} \right)}}} \leq W_{r,\lambda_{S}}$

In 710, the algorithm evaluates if the conditions shown above can bemaintained below a pre-determined threshold, e.g., 60% of the estimatedconstraint' values. If that is not achieved, in 712 the SNR of the DMDmeasurement is evaluated. Depending on this, the DMD may need to bemeasured again 704. Otherwise, in 717 it is indicated that theestimation failed. If the conditions compared in 710 are achieved, thealgorithm provides the DMD corrected pulses and the estimated EMB isobtained.

FIG. 13, shows the corrected DMD results for populations A and B. InFIG. 14 the estimated and measured bandwidths at λ_(S)=953 nm are shown.The correlation for these results is around 80%-90%.

Method 2

This method can be used to predict if the EMB at an arbitrary secondwavelength, λ_(S), is equal or greater than a specified threshold,EMB_(th), based on a DMD measurement at a different wavelength, λ_(M).As in the previous case this method utilizes features of the DMD pulsewaveforms at λ_(M), such as centroids, peak position, width, shapes,energy per radial offset, and skews. The average centroid for positionsR_(t) _(_) _(start)-R_(t) _(_) _(end) is defined using,

$\begin{matrix}{{C_{Top}\left( {R_{T\; \_ \; {start}},R_{T\; \_ \; {end}}} \right)} = \frac{\sum\limits_{r = T_{T\; \_ \; {end}}}^{R_{T\; \_ \; {end}}}C_{r\; \lambda_{M}}}{R_{T\; \_ \; {end}} - R_{T\; \_ \; {start}} + 1}} & (12)\end{matrix}$

The average centroid for positions RB_(—start)-RB_(—end) is definedusing,

$\begin{matrix}{{C_{Bottom}\left( {R_{B\; \_ \; {start}},R_{B\; \_ \; {end}}} \right)} = \frac{\sum\limits_{r = R_{B\; \_ \; {end}}}^{R_{B\; \_ \; {end}}}C_{r,\lambda_{M}}}{R_{B\; \_ \; {end}} - R_{B\; \_ \; {start}} + 1}} & (13)\end{matrix}$

A function denominated, P-Shift is computed as

P-Shift(R_(T) _(_) _(start),R_(T) _(_) _(end),R_(B) _(_) _(start),R_(B)_(_) _(end))=C_(Top)(R_(T) _(_) _(start),R_(T) _(_)_(end))−C_(Bottom)(R_(B) _(—start) ,R_(B) _(—end) )   (14)

The slopes using the peak pulse position for two or more radial regionsare computed as shown in equation below.

$\begin{matrix}{{P - {Slope\_ R}_{k}} = {\frac{1}{L}\frac{\sum\limits_{r = {R\; \_ \; {start}_{k}}}^{R\; \_ \; {end}_{k}}{\left( {P_{r\; \lambda_{m}} - T_{k}} \right)\left( {r - {\left( {{R\_ end}_{k} + {R\_ start}_{k}} \right)/2}} \right)}}{\sum\limits_{r = {R\; \_ \; {start}_{k}}}^{R\; \_ \; {end}_{k}}\left( {r - {\left( {{R\_ end}_{k} + {R\_ start}_{k}} \right)/2}} \right)^{2}}}} & (15)\end{matrix}$

where k is the index that represent the selected radial offset regionsand

$\begin{matrix}{T_{k} = {\frac{1}{{R\_ end}_{k} - {R\_ start}_{k} + 1}{\sum\limits_{r = {R\; \_ \; {start}_{k}}}^{R\; \_ \; {end}_{k}}P_{r\; \lambda_{m}}}}} & (16)\end{matrix}$

The widths for the same k regions that are computed using:

$\begin{matrix}{{M - {Width}_{k}} = {\frac{1}{L}\left\lbrack {\max_{r = {R\; \_ \; {{start}_{k}:R_{{end}_{k}}}}}\left( {W(r)} \right)} \right\rbrack}} & (17)\end{matrix}$

It should be noted for features described in (15-17), the k index cantake values from 1 to N_(k) where N_(k)<25 r of radial offsets, i.e. 25.In practice, as shown in the algorithm training example described below,low values for Nk, i.e. N_(k)=2, are enough to provide estimations withlow uncertainty.

The training method, which is described below, utilize machine learningtechniques to find the radial-offset regions that maximize thedifference between parameters such as P_shift, P_slopes and M_widths fortwo or more population of fibers. One population of fiber will haveEMB>EMB_(th) at λ_(S) and other populations will not satisfy thisconstraint. After training the estimation method simply evaluates if theextracted features from MMF under test belong to the regions foundduring training that satisfy the condition, EMB>EMB_(th) at λ_(S) basedon the DMD measurements at λ_(M).

Training for Method 2

The training process is identical to the one shown in FIG. 6, withexception of steps 606 and 608. We use the same example, starting from606 of FIG. 6, to illustrate the training method.

In step 606, the main features of the DMD pulses at λ_(M), areextracted. Note the differences with the first method which require thecomputation of the features at each wavelength, λ_(M) and λ_(S). Theextracted features are C_(r,λ) _(M) , P_(r, λ) _(M) and W_(r,λ) _(M)(centroid, peak and width) using equations (1-5).

In 608, the training is performed. The training is an iterative processthat has the goal to maximize a metric or a series of metrics thatrepresents the differences in features of two groups of fibers. Onegroup, Group 1 are composed by the MMFs that have EMB>EMB_(th) at λ_(S)and the other group, Group 2 by MMFs that have EMB<EMB_(th) at λ_(S).

Initially, all the MMFs used for training are mapped in a space definedby the P_shift, P_slopes and M_widths. The initial values of the regionsutilized in (12-17) which are {R_(B) _(_) _(start),R_(B) _(—end) },{R_(T) _(_) _(start),R_(T) _(_) _(end)}, {R_start_(k),R_end_(k)} are setto random values.

In this example, the utilized metric is a function implemented in C,Python, or Matlab, which computes p-norm distances in the mentionedspace, among the MMFs that belong to the groups Group 1 and Group 2.

$\begin{matrix}{{{M\begin{pmatrix}{R_{T\; \_ \; {start}},R_{T\; \_ \; {end}},R_{B\; \_ \; {start}},{R\_ start}_{1},} \\{{R\_ end}_{1},\ldots \mspace{14mu},{R\_ start}_{N_{k}},{R\_ end}_{N_{k}}}\end{pmatrix}} = \begin{Bmatrix}{\left( {{{P\_ Shift}{\_ Group1}} - {{P\_ Shift}{\_ Group2}}} \right)^{p} +} \\\begin{matrix}{\sum\limits_{k = 1}^{N_{k}}{{A_{1,k}\left( {{{P\_ Slopes}{\_ Group1}_{k}} - {{P\_ Slopes}{\_ Group2}_{k}}} \right)}^{p}++}} \\{\sum\limits_{k = 1}^{N_{k}}{A_{2,k}\left( {{{M\_ Width}{\_ Group1}_{k}} - {{M\_ Width1}{\_ Group2}_{k}}} \right)}^{p}}\end{matrix}\end{Bmatrix}^{1/p}},} & (18)\end{matrix}$

where A_(1,k), A_(2,k) are weight parameters to quantify the relativeimportance of each features and/or radial offset regions.

In each iteration the coordinate axes are modified by changing thevalues of {R_(B) _(_) _(start), R_(B) _(_) _(end)},{R_(T) _(_)_(start),R_(T) _(_) _(end)}, and the set of k parameters{R_start_(k),R_end_(k)}. In addition, the norm parameter p and theweights, can be also optimized in each iteration. During theoptimization process, the values can be changed at random, or indeterministic ways. For example, using the random search algorithms orusing gradient methods. The features are recomputed using (12-17) foreach new set of regions. The MMFs are mapped in the new space and theutilized metric, i.e. equation (18) is computed. The process continueuntil the metric is maximized, or until an exhaustive search isproduced.

To illustrate how the algorithm improves the metric in each iteration weuse a set of 35 MMFs. For sake of simplicity we utilize N_(k)=2,A_(1,1)−A_(1,2)−1, and p=1 and the following simplified version of themetric, (18)

$\begin{matrix}{{M\left( {R_{T\; \_ \; {start}},R_{T\; \_ \; {end}},R_{B\; \_ \; {end}},{R\_ start}_{1},{R\_ start}_{2},{R\_ end}_{2}} \right)} = \left\{ {\left( {{{P\_ Shift}{\_ Group1}} - {{P\_ Shift}{\_ Group2}}} \right)^{p} + {\sum\limits_{k = 1}^{2}{A_{1,k}\left( {{{P\_ Slopes}{\_ Group1}_{k}} - {{P\_ Slopes}{\_ Group2}_{k}}} \right)}}} \right\}} & (19)\end{matrix}$

FIG. 15 shows initial mapping of the population for one to 8000iterations. In the figure. the square markers represent MMF from Group 1and the circle markers represents MMFs from Group 2. It can be observedthat for the initial iterations 1-5000, FIGS. 15(a), (b), (c), (d) and(e) it is not possible to differentiate between both populations. After7000 iterations the algorithm capable of separate MMF from Group 1 andGroup 2. The boundaries between the groups, Group 1 and Group 2, in theplane shown in FIG. 15(i) can be established (see black trace). Based onthis classification the optimum radial offset that optimizes the featureextraction from the DMD pulse waveforms was found. The values of thefound regions are: 2 to 10 micron radial offsets for the first P-Slope(k=1), 12 to 23 microns for the second P-Slope (k=2). For the P-shiftcalculation shown in (14), the optimum regions were 2 to 3 microns forC_Top and 18-24 microns for the C_Bottom

The training using the disclosed algorithm demonstrates that the MMFsfor Group 1 and Group 2 have distinctive features that can be observedwhen the optimum set of radial regions to represent them are selected.These results demonstrate a method to predict if EMB>EMB_(th) at λ_(S)based on the DMD measurements at λ_(M).

Estimation Method

During training the optimum radial-offset regions to extract thefeatures that optimally represent MMFs that have EMB_(S)>EMB_(th) atλ_(S) were found. In the feature-space, see for example FIG. 15(i), theGroups of MMFs that have the desired characteristics can be separated bya line or in general by a polynomial that isolate two regions one forGroup 1 and another for Group 2. For the estimation process the featuresof a MMF are extracted from DMD measurements at λ_(M) and mapped in thefeature-space. If the MMF belongs to the desired regions that produceEMB>EMB_(th) at λ_(S) (see FIG. 17 the fiber is accepted. Otherwise thefiber is rejected.

Note that while this invention has been described in terms of severalembodiments, these embodiments are non-limiting (regardless of whetherthey have been labeled as exemplary or not), and there are alterations,permutations, and equivalents, which fall within the scope of thisinvention. Additionally, the described embodiments should not beinterpreted as mutually exclusive, and should instead be understood aspotentially combinable if such combinations are permissive. It shouldalso be noted that there are many alternative ways of implementing themethods and apparatuses of the present invention. It is thereforeintended that claims that may follow be interpreted as including allsuch alterations, permutations, and equivalents as fall within the truespirit and scope of the present invention.

Also note that nothing in this disclosure should be considered aslimiting and all instances of the invention described herein should beconsidered exemplary.

1. A method for estimating modal bandwidth of multimode fibers at a setof wavelengths, based on a DMD measurement performed at only onewavelength λ_(M) comprising performing a DMD measurement at a firstwavelength, extracting at least one signal feature, the signal featurebring at least one of a centroid, peak power, pulse width, and skew ofthe DMD measurement at the first wavelength, and predicting an at leastone signal feature for a second wavelength based upon the at least onesignal feature of the first wavelength, and using the predicted at leastone signal feature of the second wavelength to estimate a modalbandwidth.
 2. A method for estimating the centroids and peaks ofmultimode fibers at a set of wavelengths, based on a DMD measurementperformed at only one wavelength λ_(M).
 3. A method for estimating theDMD pulses energy position and width of multimode fibers at a set ofwavelengths, based on a DMD measurement performed at only one wavelengthλ_(M).
 4. A method to extract features of DMD pulses that describespectral dependence of modal dispersion.
 5. A method for enablingmachine learning of multimode fiber by identifying features andradial-offset regions that relates the EMB at two wavelengths.
 6. Amethod for predicting if EMB at an arbitrary wavelength, λ_(S), is equalor greater than a specified bandwidth threshold, EMB_(th), based on aDMD measurement at a different wavelength, λ_(M).
 7. A method accordingto 1 to reduce the testing time for fibers designed to operate at morethan one wavelength.
 8. A method according to 2 to evaluatemodal-chromatic dispersion properties of a MMF at multiple wavelengthbased on measurements at only one wavelength.
 9. A method according to 3to predict differential mode delay, minimum effective modal bandwidth atmore than one wavelength with reduced testing time for fibers designedto operate at more than one wavelength.
 10. A method according to claim1-4 to select MMFs with alpha parameters lower than the alpha optimumfunction at a desired spectral range.
 11. A method according to claim1-4 to select MMFs that produce modal-chromatic dispersion compensationat a desired spectral range.
 12. A method according to claim 1-4 toselect MMF that have desired modal bandwidth at desired spectral range,i.e 840nm 980 nm.
 13. A method according to claim 1-4 to sort productionin order to select standard compliant wideband fibers capable to operatein a desired spectral range, i.e. 840 nm to 1000 nm, without the need tomeasure the modal bandwidth at all the specified wavelengths.
 14. Amethod according to claim 5-6 to sort production in order to selectstandard compliant wideband fibers capable to operate in a desiredspectral range, i.e. 840 nm to 1000 nm, without the need to measure themodal bandwidth at all the specified wavelengths.
 15. A method accordingto claim 1-6 to reduce the number of modal bandwidth measurements duringmanufacturing or quality test for MMFs.
 16. A method according to claim1-6 to design MMF that complies with modal bandwidth requirements atspecific wavelengths based on the extracted features such as centroid,width, peaks, skews.